Derivative of a product and derivative of quotient of functions theorem: I don't understand its proof

I'm studying for math exam and one of the questions that often appears is related to derivative of a product of two functions.

The theorem says that $(f(x)g(x))'=f'(x)g(x)+f(x)g'(x)$.

The proof goes like this: $f(x+h)g(x+h)-f(x)g(x)=(f(x+h)-f(x))g(x)+(g(x+h)-g(x))f(x)$

After that we divide the equation by h and let h approach 0.

Now what I don't understand is how they got the right side of the proof.

Same problem with quotient:

Theorem says: $\left(\frac{f(x)} {g(x)}\right)'=\frac{f'(x)g(x)-f(x)g'(x)} {g^2(x)}$

The above comes from $\frac {f(x+h)} {g(x+h)} - \frac {f(x)} {g(x)} = \frac{(f(x+h)-f(x))g(x)-(g(x+6h)-g(x))f(x)} {g(x+h)g(x)}$

I can see from where $f(x+h)g(x)$ comes, but I can't see from where $f(x)g(x)$ came.

Solution 1:

The derivative product rule is a special case of the congruence product rule for rings, i.e.

Product Rule $\rm\quad\ a\equiv a_1,\; b\equiv b_1 \;\Rightarrow\;\; ab \equiv a_1 b_1$

Proof: $\rm\quad\; ab-a_1 b_1 \equiv a(b-b_1)+(a-a_1)b_1 \equiv 0 \quad$ QED

Thus $\rm\ \ f(x+t) \equiv \: f(x) + f'(x) \; t \;\pmod {t^2}$

and $\rm\ \ \ \,g(x+t) \equiv g(x) + g'(x) \; t \;\pmod {t^2}$

$\rm\ \ \Rightarrow\ \ f(x+t)g(x+t) \:\equiv\: f(x)g(x) + (f'(x)g(x) + f(x)g'(x)) \; t \:\;\pmod {t^2}$

$\rm\displaystyle\ \Rightarrow\ \ \frac{f(x+t)g(x+t)\: - \:f(x)g(x)}{t} \equiv\: f'(x)g(x) + f(x)g'(x) \quad\:\pmod t$

In fact this is how one universally defines derivatives in formal polynomial rings $\rm R[x]$, e.g. see here. This yields a purely algebraic approach to polynomial derivatives - devoid of limits or other topological notions.

The ring $\rm R[t]/t^2 \;$ is known as the algebra of dual numbers over the ring $\rm R$. This ring and its higher order analogs $\;\rm R[t]/t^n \;$ prove quite useful when studying (higher) derivations algebraically since they provide convenient algebraic models of tangent / jet spaces. E.g. as above, they permit easy transfer of properties of homomorphisms to derivations -- see for example section 8.15 in Jacobson, Basic Algebra II.

Dual numbers have been applied in many contexts, e.g. deformation theory [2], numerical analysis [3] (along with Levi-Civita fields), where they're viewed simply as truncated Taylor power series, and also in Synthetic Differential Geometry (SDG) [1], another rigorization of infinitesimals based on work of Lawvere and Kock. Note that SDG employs these nilpotent infinitesimals, unlike Abraham Robinson's nonstandard analysis, which employs invertible infinitesimals (hence contains infinite elements).

[1] Bell, J. L. Infinitesimals. Synthese 75 (1988) #3, 285--315.
http://www.jstor.org/stable/20116534

[2] Szendroi, B. The unbearable lightness of deformation theory, a tutorial introduction.
http://people.maths.ox.ac.uk/szendroi/defth.pdf

[3] M. Berz, Differential Algebraic Techniques,
in "Handbook of Accelerator Physics and Engineering, M. Tigner, A.Chao (Eds.)" (World Scientific, 1998)
http://bt.pa.msu.edu/cgi-bin/display.pl?name=dahape
http://bt.pa.msu.edu/NA/
http://bt.pa.msu.edu/pub/papers/

Solution 2:

There is a more comprehensible (I hope) proof coming from thinking of derivative as a linear approximation.

By the definition of derivative¹, $f(x)=f(x_0)+f'(x_0)(x-x_0)+o(x-x_0)$ and $g(x)=g(x_0)+g'(x_0)(x-x_0)+o(x-x_0)$. Multiplying the equalities we get \begin{multline} f(x)g(x)=f(x_0)g(x_0)+(f'(x_0)g(x_0)+f(x_0)g'(x_0))(x-x_0)+\\ +f'(x_0)g'(x_0)(x-x_0)^2+o(x-x_0) \end{multline} and since $f'(x_0)g'(x_0)(x-x_0)^2=o(x-x_0)$, we can rewrite it as \[(fg)(x)=(fg)(x_0)+(f'g+fg')(x_0)\cdot(x-x_0)+o(x-x_o)\] — which exactly means that (fg)'=f'g+fg'.

¹ $\lim_{x\to x_0}g(x)=a$ iff $g(x)=a+o(1)$, hence $\lim_{x\to x_0}\frac{f(x)-f(x_0)}{x-x_0}=f'(x_0)$ iff $\frac{f(x)-f(x_0)}{x-x_0}=f'(x_0)+o(1)$ which can be rewritten as $f(x)-f(x_0)=f'(x_0)(x-x_0)+o(x-x_0)$ (see e.g. wikipedia if "o(1)" notation is unfamiliar to you).

Solution 3:

An equivalent way to state the product rule is $\frac{(fg)'}{fg} = \frac{f'}{f} + \frac{g'}{g}$. I prefer this statement because it is more intuitive: it says precisely that the relative instantaneous change in $fg$ is the sum of the relative instantaneous change in $f$ and the relative instantaneous change in $g$. In other words, this is just an expression of the approximation $(1 + a)(1 + b) \approx 1 + a + b$ when $a, b$ are both small ("multiplication near the identity is addition"). In other other words, this is an expression of the familiar fact that if you increase something by 5%, then by 5% again, the total increase is just a little more than 10%.

As for turning this into a formal proof, divide both sides of the expression you're confused about by $fg$. (This is not strictly valid if $f$ or $g$ is equal to zero, but I'm going for clarity here.)

Solution 4:

This answer is really very similar to the others, but I hope that the emphasis is sufficiently different to be worth posting.

Fix a number $x$. The definition of the derivative of $f$ is that $$f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}.$$ Let's rewrite this as $$f'(x)=\lim_{h\to 0}F(h)$$ where we define $$F(h)=\frac{f(x+h)-f(x)}{h}$$ for $h\ne0$. We can rewrite this definition as $$f(x+h)=f(x)+hF(h).$$ For the same reason, $$g(x+h)=g(x)+hG(h)$$ where $$G(h)=\frac{g(x+h)-g(x)}{h}$$ and also $$g'(x)=\lim_{h\to 0}G(h).$$

We now need to investigate the derivative of $f(x)g(x)$. Now by definition this is $$\lim_{h\to0}\frac{f(x+h)g(x+h)-f(x)g(x)}{h}.$$ Manipulating this quotient gives $$\frac{f(x+h)g(x+h)-f(x)g(x)}{h}=\frac{(f(x)+hF(h))(g(x)+hG(h))-f(x)g(x)}{h}$$ $$=\frac{hf(x)G(h)+hF(h)g(x)+h^2F(h)G(h)}{h}=f(x)G(h)+F(h)g(x)+hF(h)G(h).$$ Now this yields to the "algebra of limits": using $\lim_{h\to 0}F(h)=f'(x)$ and $\lim_{h\to 0}G(h)=g'(x)$ we get $$\lim_{h\to0}\frac{f(x+h)g(x+h)-f(x)g(x)}{h}=f(x)g'(x)+f'(x)g(x)+0f'(x)g'(x)$$ which simplifies to $f(x)g'(x)+f'(x)g(x)$ which is exactly what you want.

Actually this argument is just the same as your original but expressed in a more pedestrian way, without any cunning sleights of hand. As follow-up a good exercise is to obtain the quotient rule in a similar fashion.

Solution 5:

The following is an attempt at an interpolation between (among?) the answers of Grigory M., Pierre-Yves Gaillard and Bill Dubuque (the last of which I confess that I do not completely understand, but this is one possible interpretation of it).

Let $J$ be an open interval in $\mathbb{R}$, and let $x_0 \in J$. Following PYG's suggestion, let me begin by stating exactly what I will prove: if $f,g: J \rightarrow \mathbb{R}$ are differentiable at $x_0$, then so is their product, and

$(fg)'(x_0) = f'(x_0)g(x_0) + f(x_0) g'(x_0)$.

I will prove this "congruentially", as follows:

let $R$ be the ring of all functions $f: J \rightarrow \mathbb{R}$ which are continuous at $x_0$, under the operations of pointwise addition and multiplication. Inside $R$, consider the set $I$ of all functions $f$ such that

$\lim_{x \rightarrow x_0} \frac{f(x)}{x-x_0} = 0$.

I claim that $I$ is an ideal of $R$. This is easy to prove, but I note that it makes use of the fact that every element of $R$ is continuous at $x_0$, hence bounded near $x_0$. Now:

1) For $f \in R$, $f$ lies in $I$ iff: $f(x_0) = 0$, $f$ is differentiable at $x_0$ and $f'(x_0) = 0$.

2) For $f \in R$, $f$ is differentiable at $x_0$ iff there exists $A \in \mathbb{R}$ such that $f \equiv f(x_0) + A(x-x_0)$. If so, then necessarily $A = f'(x_0)$; in particular, it is uniquely determined.

3) Thus, if $f$ and $g$ are both differentiable at $x_0$, then

$fg \equiv (f(x_0) + f'(x_0)(x-x_0))(g(x_0) + g'(x_0)(x-x_0))$

$\equiv f(x_0)g(x_0) + (f'(x_0)g(x_0) + f(x_0)g'(x_0))(x-x_0) + f'(x_0)g'(x_0)(x-x_0)^2$

$\equiv f(x_0)g(x_0) + (f'(x_0)g(x_0) + f(x_0)g'(x_0))(x-x_0) \pmod I$.

Using 2), it follows that $fg$ is differentiable at $x_0$ and

$(fg)'(x_0) = f'(x_0)g(x_0) + f(x_0)g'(x_0)$.